### Abstract

We derive analytically the spectrum for the Schrödinger equation for quasiperiodic systems with two length scales: one large macroscopic scale [e.g., a cos(2x/≫)] and one small microscopic scale [e.g., v cos(2x)]. The phase diagram includes regimes with exponentially narrow gaps due to the slowly varying potential, regimes where the rapidly varying potential amplifies these narrow gaps, and regimes with exponentially narrow Landau bands. The full devils-staircase spectrum with gaps at wave vectors q=mn/≫ develops in a hierarchical manner as a increases. The results apply to systems with superlattices, to celestial orbits with two periodic perturbations, to systems with slowly varying lattice distortions, and, in particular, to quasi-one-dimensional magnets such as bis(tetramethyltetraselenafulvalene) perchlorate [(TMTSF)2ClO4] in magnetic fields, where our findings may provide insight into the experimentally observed cascade of phase transitions.

Original language | English (US) |
---|---|

Pages (from-to) | 1392-1402 |

Number of pages | 11 |

Journal | Physical Review A |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - 1986 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Atomic and Molecular Physics, and Optics

### Cite this

*Physical Review A*,

*34*(2), 1392-1402. https://doi.org/10.1103/PhysRevA.34.1392

**Spectra and gap amplification for systems with two widely different incommensurate periodicities.** / Azbel, M. Ya; Bak, Per; Chaikin, P. M.

Research output: Contribution to journal › Article

*Physical Review A*, vol. 34, no. 2, pp. 1392-1402. https://doi.org/10.1103/PhysRevA.34.1392

}

TY - JOUR

T1 - Spectra and gap amplification for systems with two widely different incommensurate periodicities

AU - Azbel, M. Ya

AU - Bak, Per

AU - Chaikin, P. M.

PY - 1986

Y1 - 1986

N2 - We derive analytically the spectrum for the Schrödinger equation for quasiperiodic systems with two length scales: one large macroscopic scale [e.g., a cos(2x/≫)] and one small microscopic scale [e.g., v cos(2x)]. The phase diagram includes regimes with exponentially narrow gaps due to the slowly varying potential, regimes where the rapidly varying potential amplifies these narrow gaps, and regimes with exponentially narrow Landau bands. The full devils-staircase spectrum with gaps at wave vectors q=mn/≫ develops in a hierarchical manner as a increases. The results apply to systems with superlattices, to celestial orbits with two periodic perturbations, to systems with slowly varying lattice distortions, and, in particular, to quasi-one-dimensional magnets such as bis(tetramethyltetraselenafulvalene) perchlorate [(TMTSF)2ClO4] in magnetic fields, where our findings may provide insight into the experimentally observed cascade of phase transitions.

AB - We derive analytically the spectrum for the Schrödinger equation for quasiperiodic systems with two length scales: one large macroscopic scale [e.g., a cos(2x/≫)] and one small microscopic scale [e.g., v cos(2x)]. The phase diagram includes regimes with exponentially narrow gaps due to the slowly varying potential, regimes where the rapidly varying potential amplifies these narrow gaps, and regimes with exponentially narrow Landau bands. The full devils-staircase spectrum with gaps at wave vectors q=mn/≫ develops in a hierarchical manner as a increases. The results apply to systems with superlattices, to celestial orbits with two periodic perturbations, to systems with slowly varying lattice distortions, and, in particular, to quasi-one-dimensional magnets such as bis(tetramethyltetraselenafulvalene) perchlorate [(TMTSF)2ClO4] in magnetic fields, where our findings may provide insight into the experimentally observed cascade of phase transitions.

UR - http://www.scopus.com/inward/record.url?scp=35949025839&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35949025839&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.34.1392

DO - 10.1103/PhysRevA.34.1392

M3 - Article

VL - 34

SP - 1392

EP - 1402

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 2

ER -